Solutions of the Mathieu–Hill Equation for a Trapped-Ion Harmonic Oscillator—A Qualitative Discussion

We investigate solutions of the classical Mathieu–Hill (MH) equation that characterizes the dynamics of trapped ions. The analytical model we introduce demonstrates the equations of motion are equivalent to those of a harmonic oscillator (HO). Two independent approaches are used, based on two classes of complex solutions of the MH equation. This paper addresses both a damped HO and parametric oscillator (PO) for an ion confined in an electrodynamic (Paul) trap, along with stability and instability regions for the associated periodic orbits.